Understanding Eigenvalues in Rotational Transformations: A False Assertion

[DanielFaraday]

Homework Statement

True/False

If T_theta is a rotation of the Euclidean plane R² counterclockwise through an angle theta, then T can be represented by an orthogonal matrix P whose eigenvalues are lambda₁ = 1 and lambda₂ = -1.

Homework Equations

The Attempt at a Solution

Just checking to see if my thinking is right. I say false because the representation of T in orthogonal coordinates would require a transformation requiring trigonometric functions. This wouldn't be a linear transformation and therefore cannot be treated as an eigensystem.

Yes? No?

 

[Office_Shredder]

The answer is false, but I can't figure out what you're trying to say for your reasoning. Rotations are linear functions of the plane, and can be represented my matrices. You can either consider determinants, or try to demonstrate no rotation can do what is proposed to the two eigenvectors

 

[Pengwuino]

Using sines and cosines is still a linear transformation.

Think of it this way, what is the determinant of a matrix such that it's eigenvalues are -1,1?